Something they tout as an answer to every single social issue in existence. Such reasoning is fallacious, however, because in actuality life is much more complicated..
Conservative: If people don't want babies, they shouldn't have sex. That way, abortion wouldn't even be necessary. Think, liberals, think! Personal responsibility, people!
Liberal/Moderate: Ok, so if abortion was banned, it's true that less people would have irresponsible sex, and there might be far fewer abortions. You're right on that one. But what about in cases of rape? You can't mount the "personal responsibility" defense there - people don't choose to get raped.
Conservative: Make abortion legal only in cases of rape then.
L/M: Well how are you supposed to know when the hell she's lying then? You'd be encouraging women to lie. Same problem as before, she just has to employ a different method of getting around it. How are you supposed to distinguish between the liars and the true victims?
Conservative: Polygraph.
L/M: In a perfect world, those things are fully reliable. But they're clearly not. Since we don't live in a perfect world, and polygraphs sometimes fail, we may as well legalize abortion considering that there are many cases where it's very necessary. Of course there are whores who will use it as birth control. But that's unavoidable.

"personal responsibility card"

Personal responsibility is important but it doesn't take into account factors that are outside of the person's control. For instance, conservatives are operating under the assumption that welfare discourages the work ethic and personal responsibility. But what about people with Parkinson's, or similar diseases? What about valuable workers that have been laid off en masse by struggling companies and need aid to get that back on their feet? Are they necessarily lazy and lacking personal responsibility?